Class choice and the surprising weakness of Kelley-Morse set theory

Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $φ(x,X)$, then there is a class $Z\subseteq V\times V$ for which $φ(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $φ$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the Łoś theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $Σ^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall α{<}δ ψ(α,X)$ is not always provably equivalent to a $Σ^1_1$ assertion when $ψ$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.

💡 Research Summary

The paper investigates the foundational robustness of Kelley‑Morse set theory (KM), a widely accepted second‑order set theory that extends ZFC with full second‑order comprehension and a global choice function. Despite its strength, the authors demonstrate three fundamental shortcomings of KM that undermine its suitability as a foundational framework.

First, KM fails to prove the Class Choice Scheme. The scheme asserts that whenever every set x has a class X satisfying a formula φ(x,X), there exists a single class Z⊆V×V that uniformly selects a witness Zₓ for each x, i.e., ∀x φ(x,Zₓ). While GBC (Gödel‑Bernays set theory) already cannot prove this scheme, it was expected that KM, with its stronger comprehension, would. The authors construct models of KM (assuming the existence of a Mahlo cardinal) in which even very low‑complexity instances—Σ⁰₁ formulas—of the scheme fail for ω‑indexed families. Moreover, they show that the weaker set‑indexed and parameter‑free versions of the scheme also do not follow from KM; each can be made to hold while the full scheme fails, using separate model constructions. Thus KM lacks a genuine class‑level choice principle.

Second, KM does not satisfy a Łoś theorem scheme for internal second‑order ultrapowers. The authors examine ultrapowers formed inside a KM model using an ultrafilter on ω and, more dramatically, ultrapowers derived from a normal measure on a measurable cardinal. In both cases they produce KM models where the resulting internal ultrapower is not a model of KM, and where the ultrapower embedding fails to be elementary for the full second‑order language. Consequently, large‑cardinal embeddings that are elementary in ZFC are not guaranteed to be elementary in KM, showing a serious preservation failure.

Third, KM does not preserve Σ¹ₙ logical complexity under first‑order quantification. Specifically, even bounded first‑order quantifiers can raise the complexity: there are Σ¹₁ formulas ψ(α,X) such that the bounded universal statement ∀α<δ ψ(α,X) is not equivalent (over KM) to any Σ¹₁ formula. This indicates that KM cannot maintain the usual hierarchy of second‑order definability when first‑order quantifiers are introduced, contrary to expectations from second‑order arithmetic.

To remedy these deficiencies, the authors propose KM⁺, the theory obtained by adding the full Class Choice Scheme as an axiom to KM. They argue that KM⁺ immediately resolves all three issues: the scheme itself holds, internal ultrapowers become elementary (the Łoś theorem is restored), and Σ¹ₙ formulas become closed under bounded first‑order quantifiers. Moreover, citing a result of Marek, they note that KM⁺ is bi‑interpretable with ZFC⁻ (ZFC without the Power Set axiom) together with the assertion that there is a largest cardinal which is inaccessible. This places KM⁺ at a well‑understood consistency strength while providing the desired choice and preservation properties.

In summary, the paper reveals that the conventional view of KM as a robust foundation for second‑order set theory is misplaced. By exposing failures of class choice, ultrapower elementarity, and complexity preservation, the authors make a compelling case for adopting KM⁺ as a more reliable foundational system for work involving classes, large cardinals, and second‑order constructions.